Jumping to Conclusions

By: Shaila Murthy

What would happen if everyone on Earth gathered in one spot and all jumped at the same time? I can honestly say I have never pondered this question before...until today, that is. The first matter that needs to be resolved would be where we would all gather. It seems that the general consensus (from the sources I found) would be Los Angeles, where we could all fit if we stood shoulder to shoulder. Now that we've figured that out, let's get a few variables established:

Number of people on Earth: about 7.4 billion people

Average weight of said people: about 50 kg (since we are accounting for all ages)

Weight of the Earth: 6*10^24

Weight of People (based on above values): about 370 billion kg

Average height a person can jump: 1 ft./0.3 m

Gravity: 9.8 m/s

We start off with our basic equations of momentum and energy. Since everything is done within the system (we are neglecting any forces outside of the Earth), momentum and energy are conserved.

 From this, as you can see from my little note, we need to eventually find the velocity of the Earth, as that is the recoil speed at the moment of the jump. Following my handy-dandy source, I then found the velocity of the Earth from my momentum equation, using y as the direction of travel.

Now that I had this, I went on to find the velocity of the people from my energy equation (you'll see why I left it squared in the next step).

Now I have two equations, however both include the final value (i.e. the velocity of the subject) within them, and substitution here would not look pretty nor would it be very nice to solve for. Here I did wonder what would happen if I were to reverse what I did, so I would solve momentum for velocity of the people and energy for velocity of the earth, and then plug in. However, I once again encountered the issue of my desired variable being within the equation as well:

After realizing that my idea would not work, I continued to follow my source and squared Earth's velocity and plugged in the people's velocity. Earth's velocity was squared to avoid the annoyance of having to plug a square root into the equation.

As you can see from my last two steps, the Earth's velocity was still in the equation. This time, though, I was able to factor my equation to make it so that it was only on one side. From here, I solved for the actual velocity of the Earth.

Once I had my velocity equation, I plugged it into my calculator with my values from earlier, and got my answer to be 2.46*10^-19 m/s (otherwise known as an extremely small and insignificant value). So, it is true that we would move the Earth, however only the slightest infinitesimal amount. Our weight is not nearly enough to move something the size of the Earth, which is around 16 trillion times our weight... the more you know, I suppose.

Now, for T.F. Green Airport. My initial thought was if we would be able to fit everyone in the world into the airport. I found that the airport is 1111 acres, or 4496057 m^2 by 17 m tall. I was curious that if we were to split that 17 m up into x number of floors, if we would be able to fit all of the people. I assumed the height of a floor would have to be eight feet tall on average, which is equal to about 2.5 m. Los Angeles is 503 mi^2, which is equivalent to about 1.3*10^9 m^2. Assuming this, that would mean that we would need roughly 289 floors to fit everyone, and T.F. Green could have a maximum of a whopping 6.8. Clearly that would not work out. Upon some further research, I discovered someone who considered this same problem except in Rhode Island.

Rhode Island is considerably larger than Los Angeles, however it is still a small enough space that people would not be too far apart for the jump to make a difference. The results would probably change a bit since we would not be so compact, however we would all be a lot more comfortable this way. Following the jump, everyone would (obviously) wish to return home. If T.F. Green was the only airport to be used, it would take around 1,750 years to get everyone out of the state, by which time everyone would be dead. T.F. Green served about 4 million passengers in 2017, and unless they increased that number by an extremely large amount, there's no chance of getting everyone out of there in under 1.5 thousand years. Basically, gathering everyone to jump in one spot on Earth would do almost nothing and would potentially kill everyone, making it a terrible idea from every aspect (but a great theoretical question).

Sources:

1.) Allain, Rhett. "What if Everyone Jumped?" CNMN. Wired.com, 26 Aug. 2010. Web. 25 Mar. 2018.

2.) "Everybody Jump." N.p. xkcd.com, n.d. Web. 25 Mar. 2018.

3.) My good friend google (for unit conversions)